Title: Introduction to Quadratic Functions

Grade Level: 9th-10th grade

Objective: By the end of this lesson, students will be able to define and describe quadratic functions, identify their key characteristics, and understand their real-world applications.

Materials: - Whiteboard or blackboard - Markers or chalk - Handouts with practice problems - Graphing calculators (optional)

Procedure:

1. Warm-up (5 minutes):
   • Begin the lesson by asking students if they have heard the term “quadratic function” before.
   • Encourage students to share any prior knowledge or experiences they may have had with quadratic functions.
   • Write down their responses on the board.
2. Introduction (10 minutes):
   • Define a quadratic function as a function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants.
   • Explain that the highest power of the variable (x) in a quadratic function is 2, which gives it its name.
   • Emphasize that the graph of a quadratic function is a parabola, which can open upwards or downwards depending on the value of “a.”
   • Provide examples of quadratic functions and their graphs on the board.
3. Key Characteristics (15 minutes):
   • Discuss the key characteristics of quadratic functions: a) Vertex: The highest or lowest point on the graph of a quadratic function. b) Axis of symmetry: The vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. c) Roots or zeros: The x-values where the graph intersects the x-axis. d) Discriminant: The value b^2 - 4ac, which determines the nature of the roots (real, imaginary, or equal).
   • Explain how these characteristics can be determined algebraically and graphically.
   • Provide examples and guide students through the process of finding these characteristics for given quadratic functions.
4. Real-World Applications (10 minutes):
   • Discuss real-world applications of quadratic functions, such as projectile motion, optimization problems, and modeling natural phenomena.
   • Show examples of how quadratic functions can be used to solve these types of problems.
   • Encourage students to think of other scenarios where quadratic functions might be applicable.
5. Practice Problems (15 minutes):
   • Distribute handouts with practice problems involving quadratic functions.
   • Instruct students to solve the problems individually or in pairs.
   • Circulate the classroom to provide assistance and answer any questions.
6. Review and Closure (5 minutes):
   • Review the key concepts covered in the lesson, including the definition of a quadratic function and its key characteristics.
   • Ask students to share any insights or connections they made during the lesson.
   • Summarize the importance of quadratic functions in mathematics and their real-world applications.
   • Assign homework problems related to quadratic functions to reinforce the concepts learned.

Extension Activity (optional): - If time permits, students can use graphing calculators or online graphing tools to explore the effects of changing the coefficients (a, b, and c) on the shape and position of the parabola.

Assessment: - Monitor students’ participation during class discussions and their ability to solve practice problems accurately. - Review students’ completed practice problems to assess their understanding of quadratic functions and their key characteristics.